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We will show that the same type of estimates known for the fundamental solutions for scalar parabolic equations with smooth enough coefficients hold for the first order derivatives of fundamental solution with respect to space variables of…
The Lindblad master equation can always be transformed into a first-order linear ordinary differential equation (1ODE) for the coherence vector. We pose the inverse problem: given a finite-dimensional, non-homogeneous 1ODE, does a…
This paper can be seen as an attempt of rethinking the {\em Extra-Gradient Philosophy} for solving Variational Inequality Problems. We show that the properly defined {\em Reduced Gradients} can be used instead for finding approximate…
We present in this paper a general algorithm for solving first-order formulas in particular theories called "decomposable theories". First of all, using special quantifiers, we give a formal characterization of decomposable theories and…
Here we present a very efficient method to search for Liouvillian first integrals of second order rational ordinary differential equations (rational 2ODEs). This new algorithm can be seen as an improvement to the S-function method we have…
A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with…
In this study, a recursive solution technique in conjunction with generalized integrating factors is presented and applied to address first and second order linear differential equations. This approach demonstrates practical utility in…
In this paper we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear system are…
Meromorphic solutions of autonomous nonlinear ordinary differential equations are studied. An algorithm for constructing meromorphic solutions in explicit form is presented. General expressions for meromorphic solutions (including rational,…
The aim of this paper is to give two new algorithms, which are elimination free, to find polynomial and rational solutions for a given holonomic system associated to a set of linear differential operators in the Weyl algebra D = k<x_1, ...,…
Solving diverse partial differential equations (PDEs) is fundamental in science and engineering. Large language models (LLMs) have demonstrated strong capabilities in code generation, symbolic reasoning, and tool use, but reliably solving…
New numerical algorithms based on rational functions are introduced that can solve certain Laplace and Helmholtz problems on two-dimensional domains with corners faster and more accurately than the standard methods of finite elements and…
This paper deals with the index reduction problem for the class of quasi-regular DAE systems. It is shown that any of these systems can be transformed to a generically equivalent first order DAE system consisting of a single purely…
It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a…
In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can…
We study a one-dimensional ordinary differential equation modelling optical conveyor belts, showing in particular cases of physical interest that periodic solutions exist. Moreover, under rather general assumptions it is proved that the set…
In this article I present a fast and direct method for solving several types of linear finite difference equations (FDE) with constant coefficients. The method is based on a polynomial form of the translation operator and its inverse, and…
For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a…
We provide linearizability criteria for a class of systems of third-order ordinary differential equations (ODEs) that is cubically semi-linear in the first derivative, by differentiating a system of second-order quadratically semi-linear…
The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on…